208 ON THE CONDUCTION OF HEAT IN CRYSTALS.
would be equilibrium of temperature, and the flux at any point in any direction would be equal to zero. Let then a uniform. temperature, equal and opposite to that of P, be superposed on the actual system. Then the temperature at P will be reduced to zero without any change being made in the fluxes fay fyy fz. Hence these quantities will depend, not upon the absolute temperature at P, but only on its variation in the neighbourhood of P. Since, by hypothesis, these fluxes have nothing to do with the temperatures at points situated at sensible distances from P, they may be assumed to depend only on the differential coefficients du/da, du/dy, dufdz, which define the variation of temperature in the neighbourhood of P. Since moreover different systems of temperature may be superposed, it follows that fx, fy, fz are linear functions of the three differential coefficients above written. Hence equation (2) may be put under the form
du A , d*u rv d*u „, d*u
Cp~TT = A ~r,* -f B -T-75 + C ",-fc
^ dt dx* dij* dz
.
7-/-T- / ~r & "7 , 7 / * 7 f 7 , . . .OK
dy dz dzdoi) dxdy ^ ;
where x', y', zr, have been written for x, y, z,
5. Let us now refer the solid to the rectangular axes Ox, Oy, Oz, instead of Ox\ Oy\ Oz'. Let ly m, n, be the cosines of the angles odOx, x'Oy, x'0z\ let lf, m', ri, be the same for ?/', and I", m", n"9 the same for /. Then
d __ 7 d d d
~ - — . i/ y [- ffb ~^ -p % "7 •
dx dx dy dz But we have also
x = Ix + my 4- nz,
and similar formulae hold good with respect to y' and z'. Since symbols of differentiation combine with one another according to the same laws as factors, it follows that the right-hand member of equation (3) will be transformed exactly as if the symbols of differentiation were replaced by the corresponding coordinates. Hence there exists a system of rectangular axes, namely, the principal axes of the surface,
A'x'* + By + <?Y2 + W'y'z' + ZE'z'x' + 2F'a/tf = 1 . . .(4),